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flat vector bundle ：ウィキペディア英語版

flat vector bundle

In mathematics, a vector bundle is said to be ''flat'' if it is endowed with an linear connection with vanishing curvature, ''ie.'' a flat connection.
==de Rham cohomology of a flat vector bundle==

Let

\pi:E \to X

denote a flat vector bundle, and

\nabla : \Gamma(X,E)\to \Gamma(X, \Omega_X^1\otimes E)

be the covariant derivative associated to the flat connection on E.
Let

\Omega_X^* (E) = \Omega^*_X \otimes E

denote the vector space (in fact a sheaf of modules over

\mathcal O_X

) of differential forms on ''X'' with values in ''E''. The covariant derivative defines a degree 1 endomorphism ''d'', the differential of

\Omega_X^* (E)

, and the flatness condition is equivalent to the property

d^2 = 0

.
In other words, the graded vector space

\Omega_X^* (E)

is a cochain complex. Its cohomology is called the de Rham cohomology of ''E'', or de Rham cohomology with coefficients twisted by the local coefficient system ''E''.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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